The reason why balloons take off:

The air in the balloon envelope is heated by strong gas burners above the balloon's basket:

If the air is heated, it will expand. Thus the volume of the air increases and its density decreases. Consequently the warm air is lighter than the cold air which surrounds the balloon. Buoyancy (see "buoyancy in gases") is generated. This buoyancy is counteracting the weight force of the balloon and makes the balloon take off.

It is similar with a small gas balloon from the fair: This one is not filled with hot air, but with an incombustible gas, namely helium. Helium is even lighter than hot air. So you need less volume of the gas around the balloon, which is filled with helium, to make it take off.